Standard sirens Since one obtains the chirp mass independently of the distance from the phase evolution, one can measure the luminosity) distance to a compact binary coalescence using just the GW observations if one measures both polarizations). If one is able to measure the associated redshift electromagnetically, then one can obtain a measurement of the Hubble constant without the distance ladder arguments and assumptions about uniformity of standardizability over cosmic time) invoked in using observations of Type Ia SNe. This was first proposed by Bernard Schutz Nature 323, 310 1986), and has been the subject of much work since. In particular, since GW source localization is not good, identifying possible electromagnetic counterparts is already a tricky business. One also has to worry about weak lensing for the most distant and thus most interesting) sources. We haven t shown this see the Isaacson papers but GWs propagate on a curved background along null geodesics, in the geometrical optics approximation. 1 / 15 Newtonian mass octupole and current quadrupole radiation from a quasi)circular binary We now compute the mass octupole and current quadrupole for our quasi)circular binary, obtaining I klp = µ δm m ξ klp, J kl = µ δm m ξ 2 ωẑ k ξ l, where δm := m 1 m 2. Recall that ẑ denotes the normal to the binary s orbit. The Newtonian mass octupole + current quadrupole radiation is thus given by terms at once and thrice the binary s orbital period, and depends on δm/m. More explicitly, h I klp,j kl + = µmω) δm 4r h I klp,j kl = 3µmω) 2r m sin ι[5 + cos2 ι) cos φt) 91 + cos 2 ι) cos 3φt)] δm sin ι cos ι[sin φt) 3 sin 3φt)] m We thus see that the only new piece of information we gain from these higher harmonics in the nonspinning, point mass approximation) is the binary s individual masses from the chirp mass and mass difference). And, in fact, this is all one gets from any higher harmonics in this approximation. 2 / 15
Chap. II, Sec. 1.2: Inspiral of quasi)elliptic binaries We now repeat pretty much) the entire previous subsection in the more general case of a quasi)elliptic binary. The first thing to do is thus to establish a convenient parametrization of the orbit. The orbit is still confined to a plane, by conservation of angular momentum, but now its time dependence it rather more involved. We won t give the somewhat complicated) derivations here, but will simply note that the separation r is given as a function of the true anomaly ψ by and ψ = r = a1 e2 ) 1 + e cos ψ, ma1 e 2 ) r 2 = m a 3 1 + e cos ψ) 2 1 e 2 ) 3/2, by Kepler s second law i.e., conservation of angular momentum). 3 / 15 GW luminosity We begin by noting that it is most convenient to write the traceful!) mass quadrupole moment in matrix form with coordinates in the plane of the orbit) as [. M kl = µr 2 cos 2 ] ψ sin ψ cos ψ sin ψ cos ψ sin 2, ψ and use the expression for r in terms of ψ. We then compute derivatives using the expression for ψ. The resulting expression for the instantaneous luminosity is thus L inst ψ) = 8 15 µ 2 m 3 a 5 1 e 2 ) 5 1+e cos ψ)4 [121+e cos ψ) 2 +e 2 sin 2 ψ]. Of course, this instantaneous luminosity has no meaning, as such and is possibly even misleading terminology). We really want to average this over several wavelengths of the emitted radiation and since the motion is [quasi]periodic, an average over a period suffices). 4 / 15
GW luminosity cont.) Now, this average is given by a time integral, while our expression is given in terms of ψ, but we can easily change variables using the expression for ψ, giving 2π L = 2π L inst ψ) dψ ψ ω 0 0 = 32 µ 2 m 3 [ 1 5 a 5 1 e 2 ) 7/2 1 + 73 24 e2 + 37 )] 96 e4, }{{} f e) where ω 0 = m/a 3, by Kepler 3. f e) blows up as e 1. Why? Because we have faster and faster motion near periastron as e increases. 5 / 15 Change in orbital period We can now compute the averaged) change in orbital period due to GW emission [cf. Hulse-Taylor!]. The period, P, is, of course, given in terms of the semimajor axis, a, by Kepler 3, which gives a 3 P = 2π m. We relate it to the energy using a = mµ 2 E, so we have P = const.) E) 3/2 and thus ) Ṗ P = 3 Ė P 8/3 2 E = 96 5 µm2/3 f e). 2π 6 / 15
Spectrum of radiation Spectrum of radiated power from Peters and Mathews Phys. Rev. 131, 45 tational waves 1963) from binaries with two compact objects 891 s of the inthe possible urces and a ect. 5. Our in the nth 3) 1) onents, a is icity of the positionof ve detectors for the two he GW flux 2) ning the so max]) 1/2 one 3) ) 1 d, 1 kpc called chirp ency of the gn, e)/n inaries emit an their orradiation is hus, eccenrces of GW l frequency ies n of binapopulation gn, e)/n 0.5 0.4 0.3 0.2 0.1 0 0.7 0.5 0.2 0 e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 n Amplitudes of radiation from Nelemans et al. A&A 375 890 2001) 7 / 15 Fig. 1. Scale factor of the GW strain amplitude gn, e)/n for the different harmonics Eq. 3)) for e = 0, 0.2, 0.5 and 0.7. Angular momentum loss, and evolution of eccentricity Table 1. Current birth rates ν) andmergerratesν merg) per year for Galactic disk binaries containing two compact objects and their total number #) in the Galactic disk, as calculated with the SeBa population synthesis code see text). We have Type ν ν merg # wd, wd) 2.5 10 2 1.1 10 2 1.1 10 8 [wd, wd) 3.3 3 4.2 10 7 ns, wd) 2.4 Ė 10= 32 µ 2 m 3 4 1.4 10 4 2.2 10 6 ns, ns) 5.7 10 5 2.45 10 a 5 5 7.5 10 5 bh, wd) 8.2 10 5 1.9 10 6 1.4 10 6 bh, ns) 2.6 5 2.9 10 6 4.7 10 5 bh, bh) 1.6 L 10= 32 µ 2 m 5/2 4 2.8 10 6 synthesis code SeBa Portegies Zwart & Verbunt 1996; Portegies Zwart & Yungelson 1998; Nelemans et al. 2001b). The basic assumptions used in this paper can be summarised as follows. The initial primary masses are distributed according to a power law IMF with index 2.5, the initial mass ratio distribution e 2 is = taken 1 + 2EL2 flat, the initial semi major axis distribution flat in log a up to a =10 6 R, and the eccentricities follow P e) 2e. Thefractionofbinaries in the initial population of main-sequence stars is 50% 2/3 of all stars are in binaries). A difference with other studies of the ȧ populations = 64 µm 2 of close binaries is that the mass transfer from a giant5to a main a 3 f e), sequence star of comparable mass is calculated using an angular momentum balance formalism, as described in Nelemans et al. ė = 304 µm 2 e 2000b). For the star formation rate of the Galactic disk 15 a 4 we use an exponential function: which gives, using Note SFRt) that=15 circular exp t/τ) orbits remain M circular, yr 1, which is nice... 4) where τ = 7 Gyr. With an age of the Galactic disk of 1 5 f e), 1 1 a 7/2 1 e 2 ) 2 + 78 ) e2, m 2 µ 3, a = mµ 2 E, 1 e 2 ) 5/2 1 + 121 ) 304 e2. 8 / 15
Angular momentum loss, and evolution of eccentricity cont.) Thus we have and d log e d log a = 19 1 e 2 12 1 + 73/24)e 2 + 37/94)e 4 ae) = c 0 e 12/19 1 e 2 It is therefore clear that a = 0 e = 0. It is convenient to define so ge) := e12/19 1 e 2 1 + 121 ) 870/2299 304 e2. 1 + 121 ) 870/2299 304 e2, 1 + 121 ) 304 e2, ae) = a 0 ge) ge 0 ). 9 / 15 Angular momentum loss, and evolution of eccentricity cont.) KARL MARTEL AND ERIC POISSON PHYSICAL REVIEW D 60 124008 into components that oscillate at once, twice, and three times the orbital frequency f orb 1/P. The wave spectrum is actually more complicated than this because the angular velocity d /dt is not uniform. Nevertheless, this decomposition of the waves into three components is still meaningful and useful. The detector responds to the linear combination st) f s t) f s t), where f and f are the detector s beam factors 1. Our calculations are not sensitive to the numerical value of the beam factors; we choose f 1 and f 0, so that st) s t). Similarly, our calculations are not sensitive to the numerical value of and ; we choose /4 and 0. We assume that the gravitational-wave signal began before the waves entered the frequency band of our detector. In our computations, it is sufficient to start the signal immediately before it enters this band. We shall denote by f min the From Peters, lowerphys. end ofrev. the instrument s 136, 1224 1964) frequency band; for the initial version of the LIGO detector, f min 40 Hz. The signal component that first enters the band is the one that oscillates at three times the orbital frequency. We must therefore impose 3 f orb f min to ensure that our simulated signal begins sufficiently early; an actual signal would of course begin much earlier. FromFIG. Martel 2. Plots andofpoisson, st) up to PRD an overall 60, 124008 scaling 1999) for a 1.4 1.4 binary system with initial eccentricity e 0 0.5. The main figure 2 shows 1.4M the waveform, e 0 = for 0.5its entire duration. The bottom inset shows the waveform at early times, when the eccentricity is still large. The top inset shows the waveform at late times, when the eccentricity is much reduced. Notice the monotonic increase of both the amplitude and frequency. 10 / 15
Time to coalescence It is easiest to integrate the expression for ė, using the explicit solution for ea), yielding [ 5 a 4 ] 0 48 1 e0 g 4 e)1 e 2 ) 5/2 t c = 256 m 2 µ 19 g }{{} 4 e 0 ) 0 e1 + 121 de. 304 e2 ) circular orbit result As one might expect, the coalescence time decreases rather rapidly with increasing eccentricity. 11 / 15 Time to coalescence From Peters, Phys. Rev. 136, 1224 1964) 12 / 15
An extremely brief note about GWs from sources at cosmological distances As mentioned previously, one obtains the redshift effects one would expect for propagation of GWs from cosmological distances. In the waveform: The observed frequency is given by f obs = f intrinsic /1 + z) z is the redshift). One replaces r by the luminosity distance, d L. One replaces the chirp mass M by 1 + z)m 13 / 15 Phenomenology of spin All astronomical objects spin the members of compact binaries should be no exception. And black holes, at least, can have spins that are large enough to significantly affect the orbital motion: The total angular momentum is conserved up to radiative losses), but the individual spins and the orbital angular momentum can, and do, precess, except in special cases e.g., both spins aligned with the orbital angular momentum). [This is like Lense-Thirring precession, and is due to gravitomagnetic effects.] This can lead to wild modulations in the waveform, and causes all sorts of headaches for detection. But these precessing systems are quite interesting, so it s worthwhile to look for them. 14 / 15
Phenomenology of spin cont.) 4e-19 met a taper wave 3e-19 h + t) 2e-19 1e-19 0-1e-19-2e-19-3e-19-4e-19 0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07 3e+07 From Key and Cornish, PRD 83, 083001 2011) t sec) FIG. 2. The time dependent gravitational waveform h þ ðtþ at the solar system barycenter for a binary black hole system with e 0 ¼ 0:3 Source 1 in Table II) during the final year before merger is shown. Mass ratio 2 : 1 total mass 3 10 6 M ), spins 0.5 and 0.8, initial eccentricity, 0.3. with Ou signa from times wave Our s LISA ters. I ever, mean at tim impro we fi the ot 15 / 15 083001-3